1. Field of the Invention
The present invention relates to Hammerstein models, and particularly to a computerized method for identifying multi-input multi-output (MIMO) Hammerstein models for engineering design applications.
2. Description of the Related Art
The Hammerstein Model belongs to a family of block oriented models, and is made up of a memoryless nonlinear part followed by a linear dynamic part. It has been known to effectively represent and approximate several industrial processes, such as, for example, pH neutralization processes, distillation column processes, and heat exchange processes. Hammerstein models have also been used to successfully model nonlinear filters, biological systems, water heaters, and electrical drives.
A significant amount of research has been carried out on identification of Hammerstein models. Systems can be modeled by employing either nonparametric or parametric models. Nonparametric representations involve kernel regression or expansion of series, such as the Volterra series. This results in a theoretically infinite number of model parameters, and is therefore represented in terms of curves, such as step responses or bode diagrams. Parametric representations, such as state-space models, are more compact, as they have fewer parameters and the nonlinearity is expressed as a linear combination of finite and known functions.
Development of nonlinear models is the critical step in the application of nonlinear model based control strategies. Nonlinear behavior is prominent in the dynamic behavior of physical systems. Most physical devices have nonlinear characteristics outside a limited linear range. In most chemical processes, for example, understanding the nonlinear characteristics is important for designing controllers that regulate the process. It is rather difficult, yet necessary, to select a reasonable structure for the nonlinear model to capture the process nonlinearities. The nonlinear model used for control purposes should be as simple as possible, warranting minimal computational load and, at the same time, retaining most of the nonlinear dynamic characteristics of the system. The following convention has been used in what follows: upper case variables in bold represent matrices, lower case bold variables represent vectors, and lower case regular (i.e., non-bold) variables represent scalar quantities.
Many model structures have been proposed for the identification of nonlinear systems. The nonlinear static block followed by a dynamic block in the Hammerstein structure has been found to be a simple and effective representation for capturing the dynamics of typical chemical engineering processes such as distillation columns and heat exchangers, for example. Nonlinear system identification involves the following tasks: Structure selection, including selection of suitable nonlinear model structures and the number of model parameters; input sequence design, including the determination of the input sequence u(t) which is injected into the system to generate the output sequence y(t); noise modeling, which includes the determination of the dynamic model which generates the noise input w(t); parameter estimation, which includes estimation of the remaining model parameters from the dynamic system data u(t) and y(t), and the noise input w(t); and model validation, including the comparison of system data and model predictions for data not used in model development.
Hammerstein systems can be modeled by employing either nonparametric or parametric models. Nonparametric models represent the system in terms of curves resulting from expansion of series, such as the Volterra series or kernel regression. In practice, these curves are sampled, often leading to a large number of parameters. Parametric representations, such as state-space models, are more compact and have fewer parameters, while the nonlinearity is expressed as a linear combination of finite and known functions. In parametric identification, the Hammerstein model is represented by the following set of equations:y(t)=−a1y(t−1)− . . . −any(t−n)+bov(t)+ . . . +bmv(t−m)  (1)v(t)=c1u(t)+c2u2(t)+ . . . +cLuL(t)  (2)where v(t) describes the nonlinearity, L is the order of the nonlinearity, and y(t) and u(t) are the outputs and inputs of the system.
In MIMO Hammerstein models, as noted above, a nonlinear system is represented as a nonlinear memory-less subsystem f(.), followed by a linear dynamic part. The input sequence u(t) and the output sequence y(t) are accessible to measurements, but the intermediate signal sequence v(t) is not. As shown in FIG. 1B, the static nonlinear element scales the inputs u(t) and transforms these inputs to v(t) through a nonlinear arbitrary function f(u). The dynamics of the system are modeled by a linear transfer function, whose outputs are y(t).
Many different techniques have been proposed for the black-box estimation of Hammerstein systems from input-output measurements. These techniques mainly differ in the way that static nonlinearity is represented and in the type of optimization problem that is finally obtained. In parametric approaches, the static nonlinearity is expressed in a finite number of parameters. Both iterative and non-iterative methods have been used for determination of the parameters of the static-nonlinear and linear-dynamic parts of the model. Typical techniques, however, are extremely costly in terms of computational time and energy.
Additionally, most techniques designed to deal with Hammerstein models focus purely on single-input single-output (SISO) models. Identification of MIMO systems, however, is a problem which has not been well explored. Identification based on prediction error methods (PEM), for example, is a complicated function of the system parameters, and has to be solved by iterative descent methods, which may get stuck into local minima. Further, optimization methods need an initial estimate for a canonical parametrization model; i.e. models with minimal numbers of parameters, which might not be easy to provide.
It has been shown that this minimal parametrization can lead to several problems. PEM have, therefore, inherent difficulties with MIMO system identification. More recent studies have also shown that maximum likelihood criterion results in a non-convex optimization problem in which global optimization is not guaranteed. Subspace identification methods (SIM) do not need nonlinear optimization techniques, nor do these methods need to impose to the system a canonical form. Subspace methods therefore do not suffer from the inconveniences encountered in applying PEM methods to MIMO system identification. Thus, it would be desirable to make use of this advantage, modeling the linear dynamic subsystem of the Hammerstein model with a state-space model rather than polynomial models. Thus, a method for identifying multi-input multi-output Hammerstein models solving the aforementioned problems is desired.